It is the classic Spitting Image sketch of Margaret Thatcher in her pomp. The satirists created a scene in which the rubber puppet of the prime minster sat at a restaurant table surrounded by her cabinet. She orders raw steak. The waitress then asks: “And what about the vegetables?” Motioning to her cringing ministers, Mrs T replies: “They’ll have the same as me.”

That was a caricature. In his recently published memoir, Ken Clarke contends that the “Iron Lady” liked ministers who argued back and she promoted him even though they had many humdinger rows. But the Thatcher legend remains very lively in the Tory imagination. It was played up to by Theresa May when she arrived at Number 10 just over 100 days ago. “Iron Mayden” and “the new Maggie” were among the welcoming headlines in the rightwing press. Her team didn’t object. They did not discourage anyone from portraying her as a reincarnation of the dominatrix of folklore.

In her early days at Number 10, there was an ambition to achieve a grip over government more steely than that achieved by Mrs T even at the zenith of her power. Ministers reported that they were being forbidden to make any statement or give any interview unless it had first been cleared through Number 10. I wrote at the time that Mrs May would discover that she would not be able to impose such a stifling level of control. What I did not foresee – and neither did she nor anyone else – is just how rapidly cabinet cohesion would unravel. Discipline is now breaking down in a way that Mrs T would never have tolerated.

There are almost daily reveals of confidential papers prepared for internal discussions between ministers, especially of anything touching on Brexit. Mrs May says she will not give a “running commentary” on how she plans to approach the negotiations. We don’t need one because we have a running tap of leaks from within her cabinet. These are being accompanied by a drip feed of poisonous briefings, as some ministers try to promote themselves and their ambitions at the expense of rivals they seek to thwart or damage.

At the most recent prime minister’s questions, Mr Clarke was moved to condemn the “Brexiter ministers” whom he identified as the principal culprits. He urged Mrs May to get them under control. The majority of the cabinet agree. Most of them are not incessantly leaking and relentlessly briefing. The well behaved are growing furious with the troublemakers. “Ken is right,” says one senior minister angered by the antics of some of his colleagues. “Theresa needs to grip this before it spirals out of control.” It only takes a few to make mischief. It doesn’t take long for mischief to turn into mayhem. The next stop on the line after mayhem is anarchy.

The copious leaking and nasty briefing is being accompanied by a mounting number of public rebukes by Number 10 of ministers who are deemed to have spoken out of turn. The three Brexiters – Boris Johnson, David Davis and Liam Fox – have all been give the ceremonial “slap-down”. This was the cause of some satisfaction over at the Treasury, which has had a never less than fractious relationship with the Brexiters.

Last week, it was the turn of the second most important person in the government to feel the sting of the prime minister’s displeasure. On Wednesday, the chancellor suggested that students could be excluded from the net migration target. The following day, he was publicly told off by the next-door neighbour when Number 10 declared that this was “categorically” not Mrs May’s position. These reprimands are only superficially a display of strength. If ministers were really afraid of incurring the wrath of the boss, they would not get out of line in the first place.

Some of Mrs May’s problems with managing her government are down to her style. She is famous among colleagues for what one minister calls her “long silences”. She lets other people babble on, while giving away little about what she thinks. Many of us have encountered managers who think this is a clever technique. The trouble with it as a way of running a government is that it leaves ministers without a clear guide to what the boss really wants and encourages the pursuit of individual agendas.

Tony Blair was much attacked for emasculating the cabinet as a decision-making body and making the big calls with small huddles of trusties in the prime ministerial den, what became known as “sofa government”. A similar criticism was levelled at David Cameron. Mrs May set out with the intention of restoring the traditional model of “collective cabinet responsibility”. Ministers would argue things out in private. Everyone would get an opportunity to have their say behind closed doors, but once a decision was made, all would be expected to unify around it in public. There is a lot to be said in favour of a less cliquey, more formal method of cabinet government, but it can only work if everyone can be relied on to obey the rules.

Mrs May is beginning to learn why her predecessors became so wary of having frank and full-blooded discussions in cabinet and its committees. Unless everyone around the table can be trusted to respect the confidentiality of the meeting, there is a high risk that partial accounts of their discussions will be leaked by ministers seeking to advantage themselves and undermine competitors.

Then there is a lingering legacy of the coalition years. When the Conservative and Lib Dem ministers first went into government together, they colluded in the fiction that they were all one big happy family. As time went on and tensions increased, they became more open about ventilating their differences in public. This loosened the bonds of collective responsibility. They were then shattered when David Cameron permitted his cabinet to campaign on both sides of the Brexit referendum. Ministers acquired the habit of arguing with each other on TV – and in the most vituperative terms.

It is harder for the prime minister to reassert the authority of her office when she hasn’t won an election in her own right. Some of her friends think she ought to make an example of one of the leakers and fire him, but that would be risky when she has a small majority and a large number of enemies already sitting on her backbenches.

Added to that is a change in the dynamics between Numbers 10 and 11. The most important relationship in any government is that between the prime minister and chancellor. When they form a solid axis, they can dominate the cabinet. If that relationship breaks down, all hell tends to break loose. Mrs May chose Philip Hammond as her chancellor because she has known him since they were at Oxford together and they had a decent relationship as ministers. His unflashy, deadpan style seemed suited to the challenge of soothing markets rattled by Brexit.

The duo now find themselves, to an extent neither had anticipated, on different sides of several arguments. His priority is to protect financial services, the car industry and all the other elements of the economy that will be imperilled by a bad Brexit. That angles him towards a softer version of departure than the prime minister, who places much more emphasis on control of immigration. It ought to disturb them both that they have already got to the point where he has to deny that he is contemplating resignation and she has to issue a statement expressing “full confidence” in her chancellor.

It used to be a reliable rule that divided parties don’t win elections. Tories can be forgiven for thinking that this stricture doesn’t apply to them anymore, especially in the absence of a credible threat from their external opponents. The Conservatives were obviously divided about Europe before the 2015 election and still managed to squeak a parliamentary majority. They fought each other viciously during the referendum and that battle goes on in a new form. Yet they still enjoy double-digit leads in opinion polls. This is encouraging some Tories to believe that there is no longer a penalty for being undisciplined.

Yet there are big reasons why it does still matter that the government is advertising its divisions to the world. It is hard to come to sensible decisions when the cabinet leaks like a sieve. As a writer about politics and as a citizen, I love a juicy leak, because it exposes arguments within government that would otherwise remain hidden. But I take the point that it becomes impossible for ministers to conduct candid and productive discussions if they live in constant fear that anything they say to each other in private will appear on the front page of the next day’s newspapers. Making a success of Brexit – or at least limiting the economic damage inflicted by it – will depend on fashioning a coherent and unified negotiating position and convincing the EU that Britain has a serious and trustworthy government. That is all undermined when key ministers are at war with each other. It also compounds the Brexit-induced febrility in financial markets.

As for Mrs May, she acquired her early popularity with voters by projecting herself as a steady and competent hand on the tiller. That image will be increasingly hard to sustain when the public begins to notice that her crew are at each other’s throats.